Remarks on Contact and Jacobi Geometry

Symmetry Integrability and Geometry-Methods and Applications

article

Andrew James Bruce¹  Katarzyna Grabowska²  Janusz Grabowski³ 

[1] Mathematics Research Unit, University of Luxembourg;Faculty of Physics, University of Warsaw;Institute of Mathematics, Polish Academy of Sciences

关键词: symplectic structures;    contact structures;    Poisson structures;    Jacobi structures;    principal bundles;    Lie groupoids;    symplectic groupoids;   

DOI  :  10.3842/SIGMA.2017.059

来源: National Academy of Science of Ukraine

【 摘 要 】

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures , i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

【 授权许可】

Unknown